About work and energy

Suppose we assume the concept of force and Newton second law. Suppose a constant force over a little displacement of a particle, we want to know how the quantity $v^{2}$ change (why this quantity? a sudden brain wave, I guess; or by observation):

d (v^{2}) / d t = 2 v \frac{d v}{d t} d t = 2 \frac{d x}{d t} \frac{d v}{d t}

and since $F = m \frac{d v}{d t}$ (assuming a constant mass),

d (v^{2}) / d t = 2 \frac{F}{m} \frac{d x}{d t}

and treating $F$ as constant and integrating:

Δ (v^{2}) = 2 \frac{F}{m} Δ x

\begin{matrix} (kinetic) & Δ (\frac{1}{2} m v^{2}) = F Δ x \end{matrix}

This leads us to define $W := F Δ x$ the work, and $T := \frac{1}{2} m v^{2}$ the kinetic energy, concluding that $Δ T = W$ . Interpretation: if we spend work we get kinetic energy. My opinion: this is an equivalent way to establish second Newton law. The concepts it uses are at the same level of "reality" (or unreality): force, work, energy. We are only naming specifics mathematical expressions, according to our limited intuition. The only difference is that we are in an upper "level of integration or derivability".

For some kind of forces (everyone, if we are immersed in fundamental physics), the work done displacing a particle along a curve only depends on initial and final point. If we fix an initial point $P$ , work define an scalar field on $M$ , namely

- U (P^{'}) = work done from P to P^{'}

And so

work done from P^{'} to P^{″} = U (P^{'}) - U (P^{″}) = - Δ U

We call $U$ the potential of the force $F$ and it is easy to check that

F = - \partial_{x} U

From second Newton law (multiplied by $\dot{x}$ ):

m \ddot{x} \dot{x} = - \partial_{x} U \dot{x}

Integrating:

\begin{matrix} (potencial) & Δ T = - Δ U \end{matrix}

that is to say, the conservation of energy:

T_{1} + U_{1} = T_{2} + U_{2}

Conclusion: We begin by believing Newton ideas, basically "force" idea and second Newton law. They seems intuitive (relatively, at least) because of our every day life. But if we apply maths to them, we are led to equations like (kinetic) and (potencial) that suggest that we must also put names to other quantities because they emerge in different places: $W$ , $T$ and $U$ . Although they are less intuitive that Newton original concepts, they let us formulate the same model for physics.

In other words:

Para Newton el escenario donde ocurre todo es un mundo de 3 dimensiones reales, un espacio en el sentido de Euclides. El tiempo es un "medio" gracias al cual todo va cambiando. Ahí tenemos partículas (podrían ser átomos), fuerzas, etcétera, y hay propiedades que hemos observado o deducido. De las fuerzas lo único que sabemos es que existen, pero no por qué. Se parte de tres leyes que hacen referencia al concepto de "fuerza" como un objeto vectorial. Ello da solución a muchos problemas físicos, pero a través de construcciones altamente complicadas con cálculos de vectores. El concepto de energía surge después como un "potencial" para (una capacidad de) desarrollar una "fuerza", y viene a explicar el por qué de la existencia de esta última (lo que seguiría sin quedar claro es por qué hay "potenciales"). Es decir, la energía potencial se entiende como algo cuyo gradiente origina la fuerza (campos de fuerza). And the conservation of energy substitute to Newton second law.

After this, when we are used to this more abstract concepts, when we even begin to have an intuition about what energy IS in reality, we give another jump in abstraction and go up to Lagrangian Mechanics.